Like most businesses, an auto transport service should be willing to give you references, to offer proof of insurance, and to show you any necessary accreditations. If they can't or won't, move on to the next. Remember, there are plenty of fish in the auto transport sea.
If you're planning to move to an island or an overseas country and want to keep the car you've got, you'll need to find a reliable auto transport company. You car will be shipped in a large shipping container, ideally with other cars, and carried to its new home by cargo ship.